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I) The index law (AB)' = A rB r does not hold for square matrices. In particular, (AB)2, which means ABAB, need not equal A2B2, which means AABB. (ii) In expanding products such as (A + B)( e + D), care must be taken to preserve the distinction between the matrix products XYand YX. g. one might, without thinking, rewrite (A + Bf as A 2 + 2AB + B2, but in fact this would be wrong. Correct would be (A+B)2 = (A + B)(A +B) = A(A+B)+B(A+B) = A 2 +AB+BA+B 2, which is different from A2+2AB+B2 unless A and B commute.

Using position vectors, show that B, F, E are collinear, and find the value of BF/FE. 3. Let ABC be a triangle, and let D, E, F be points on the sides BC, CA, AB, respectively, such that BD/DC = CE/EA = AF/FB. Prove that the centroids of the triangles ABC and DEF coincide. 4. Let ABCD be a quadrilateral, and let M 1, M 2, M 3, M 4 be the mid-points of the sides AB, BC, CD, DA, respectively. By using position vectors relative to an arbitrary origin, show that M 1 M 2M 3M4 is a parallelogram. Let 0 be a point not in the plane of ABCD, and let G1 , G2 , G3 , G4 be the centroids of the triangles OCD, ODA, OAB, OBC, respectively.